Question
Evaluate:
\[ \sin\left(2\cos^{-1}\left(-\frac{3}{5}\right)\right) \]
Solution
Let
\[ \theta = \cos^{-1}\left(-\frac{3}{5}\right) \Rightarrow \cos\theta = -\frac{3}{5} \]
Since \( \theta \in [0,\pi] \), angle lies in second quadrant ⇒ sinθ > 0
\[ \sin\theta = \sqrt{1 – \cos^2\theta} = \sqrt{1 – \frac{9}{25}} = \frac{4}{5} \]
Now use identity:
\[ \sin 2\theta = 2\sin\theta \cos\theta \]
\[ = 2 \cdot \frac{4}{5} \cdot \left(-\frac{3}{5}\right) = -\frac{24}{25} \]
Final Answer:
\[ \boxed{-\frac{24}{25}} \]
Key Concept
Determine the quadrant first, then apply double-angle identity carefully.